A number that represents the likelihood an event will occur.
\[p(A) = \frac {The \, number \, of \, times \, A \, can \, occur}{The \, total \, number \, of \, possible \, outcomes}\]
Probabiltity vs Proportion
Probability
Represents the chance of some event occurring
Theoretical
Event has not occurred
Proportion
Summarizes how frequently some event actually occurred
Empirical
Event has occurred
Probabiltity vs Proportion Coin Example
If we flip a fair coin, the probability that it lands on heads is 1/2 or 50%.
But if we flip a coin 20 times and count the number of times it lands on heads, lets say 12 times, then our proportion is 12/20 or 60%.
Coin Example
Let’s flip some coins!
Card Example
Discrete Probability
Discrete Probability
A type of probability that deals with the likelihood an event will occur within a finite (limited) possible outcomes.
Binomials
Trials (i.e., an act with an different outcomes) that have exactly two possible outcomes.
Coin flips
Chances of success for a free-throw shooter in basketball, where 1 = a basket made and 0 = a miss.
Binomial probability distribution contains all the possible results over a set of trials and lists the probability of each result.
The Binomial Coefficient and the Probability Distribution
Criteria
1.) Fixed Trials
2.) Independent Trials
3.) Fixed Probability of Success
4.) Two Mutuallly Exclusive Outcomes
The Binomial Coefficient and the Probability Distribution
\[ p(x) = \binom{n}{x}p^xq^{n-x}\]
p(x) = probability of x occurring
x = number of successes
n = sample size
p = probability event will occur
q = probability event will not occur
Breakdown of “n choose x”
\[\binom{n}{x} = \frac{n!}{x!(n-x)!}\]
n = sample size
x = number of successes
! = factorial
What is q?
\[q = (p - 1)\]
p = probability event will occur
q = probability event will not occur
Example
Do people prefer Qdoba or Chipotle? Let’s say asked three people what they prefer.
Example
But we are conditioning on order here.
0.5 * 0.5 * 0.5 = 0.125
0.5 * 0.5 * 0.5 = 0.125
0.5 * 0.5 * 0.5 = 0.125
0.125 + 0.125 + 0.125 = ?
Let’s work the formula
Do people choose Qdoba or Chipotle? Let’s say asked 3 people what they prefer (with a probability (p) of 0.50 for choosing Qdoba) - 2 people chose Qdoba and 1 chose Chipotle.
What is the probability that people preferred Qdoba?
\[p(x) = (\frac{n!}{x!(n-x)!})p^x(1-p)^{n-x}\] x = ?
n = ?
p = ?
Let’s attack our ‘n choose x’ first
\[\binom{n}{x} = \frac{n!}{x!(n-x)!}\]
\[\binom{3}{2} = \frac{3!}{2!(3-2)!}\]
\[\frac{6}{2}=3\]
Let’s continue
\[ p(x) = (3)p^xq^{n-x}\]
\(p^x\) is the probability that Qdoba was chosen 2 out of 3 times.
Let’s continue
\[p(x) = (3)0.5^2(1-p)^{n-x}\]
1.) This is the probability that someone will prefer Chipolte
\[(1 - 0.50) = 0.50\]
2.) This corresponds to the one person that preferred Chipotle
